Ultralight scalars and resonances in black-hole physics
aa r X i v : . [ g r- q c ] F e b Ultralight scalars and resonances in black-hole physics
Ryuichi Fujita and Vitor Cardoso , , CENTRA, Departamento de F´ısica, Instituto Superior T´ecnico – IST,Universidade de Lisboa – UL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal Perimeter Institute for Theoretical Physics Waterloo, Ontario N2J 2W9, Canada and Theoretical Physics Department, CERN, CH-1211 Gen`eve 23, Switzerland
Ultralight degrees of freedom coupled to matter lead to resonances, which can be excited when theCompton wavelength of the field equals a dynamical scale in the problem. For binaries composed ofa star orbiting a supermassive black hole, these resonances lead to a smoking-gun effect: a periastrondistance which stalls , even in the presence of gravitational-wave dissipation. This effect, also calleda floating orbit , occurs for generic equatorial but eccentric orbits and we argue that finite-size effectsare not enough to suppress it.
I. INTRODUCTION
Massive scalars or pseudoscalars are a natural exten-sion of General Relativity and have been used in thePeccei-Quinn theory or improvements thereof to solvethe strong CP problem, in scalar-tensor modifications ofgravity, as dark matter models, and in most cosmologicalmodels. They arise as moduli and coupling constants instring theory and we now know that at least one scalarexists in nature, namely the Higgs boson. For reviews,see, for example, Refs. [1, 2].Consider a massive scalar field ϕ of mass m s = ~ µ s ,coupled nonminimally to matter and described at linearorder by the Klein-Gordon equation sourced by a scalarcharge, (cid:2) (cid:3) − µ s (cid:3) ϕ = α T . (1)The source T is the trace of the stress-energy tensor and α describes a scalar charge for matter. This equationdescribes a wide range of situations [3–6].Take a dynamical situation for which the source T isa periodic function of time and describes, for example,a two-body problem of characteristic frequency Ω p . Itis then clear that, for scalars with an “eigenmode” at ω = m Ω p , a resonance will be excited by the periodicmotion, with m an integer. In Refs. [3, 4] this was shownto be correct for a system composed of a supermassive,spinning black hole (BH) and a pointlike star, driven bygravitational- and scalar-wave emission. The resonanceoccurs at ω = m Ω p ∼ µ s and leads to the following:(i) There is a surprising effect for BHs spinning abovea critical, µ s -dependent threshold. Because of su-perradiance [7], matter can hover into “floating or-bits” for which the net gravitational energy loss atinfinity is entirely provided by the BH’s rotationalenergy. Orbiting bodies remain floating until theyextract sufficient angular momentum from the BH,or until perturbations or other effects disrupt theorbit.(ii) There is a speedup in the inspiralling of the star,when the resonance is nonsuperradiant. In otherwords, for slowly spinning BHs, the orbiting star“sinks” in, once it comes across the resonance. Floating is, in this context, a nonperturbative effectdue to a resonance between the BH and the scalarfield (more precisely, induced by the massive term).Any resonant, nonperturbative phenomenon is extremely(“nonperturbatively”) sensitive to perturbations of theconditions that give rise to resonance. Several effectscould suppress the resonances, most notably eccentric or-bits, finite-size effects and conservative self-force effects.The purpose of this work is to compute accurately theresonance details for eccentric orbits, and to estimatewhether finite-size effects can destroy the resonances. Weshow that neither eccentricity nor finite-size effects seemable to affect the resonance significantly. II. SETUP
We consider a stellar-mass compact object inspirallinginto a supermassive BH. For such extreme mass ratio in-spirals (EMRIs), generic scalar-tensor theories reduce tomassive or massless Brans-Dicke theory [4] and the fieldequations for the scalar field at the first-order perturba-tion are given by Eq. (1). Our main results are, to a largeextent, independent of the source term on the right-handside, but for concreteness we focus on source terms of theform T = Z d ¯ τ p − ¯ g (0) m p δ (4) ( x − X (¯ τ )) , (2)corresponding to the trace of the stress-energy tensor of apoint particle with mass m p , where ¯ g (0) is the background(Kerr) metric and X (¯ τ ) is the orbit of the particle. A. Eccentric orbits in the equatorial plane of ablack hole
EMRIs in the eLISA band are expected to have or-bital eccentricities, of the order of ∼ . r (cid:18) drdτ (cid:19) = R ( r ) , (3) r (cid:18) dφdτ (cid:19) = a ∆ P r ( r ) − ( a E − L ) ≡ Φ( r ) , (4) r (cid:18) dtdτ (cid:19) = r + a ∆ P r ( r ) − a ( a E − L ) ≡ T ( r ) , (5)where E and L are the energy and the angular momentumof the particle, R ( r ) = [ P r ( r )] − ∆ [ r + ( a E − L ) ], P r ( r ) = E ( r + a ) − a L and ∆ = r − M r + a .Geodesics in the equatorial plane of the BH could bespecified by the constants of motion ( E , L ). For the caseof a bound orbit in the equatorial plane, one can use an-other set of orbital parameters ( p, e ), where p is a semi-latus rectum and e is an eccentricity. Using two turningpoints of the radial motion, the apastron r a and the pe-riastron r p , ( p, e ) can be defined through r p = p e , r a = p − e . (6)Solving R ( r p ) = 0 and R ( r a ) = 0 in Eq. (3), one cancompute the set of the constants of motion ( E , L ) in termsof a given set of orbital parameters ( p, e ). It is worthnoting that for a bound orbit in the equatorial plane onecan compute two fundamental frequencies, Ω r and Ω φ ,as [17]Ω r = 2 π Z r a r p dr dtdr ! − , Ω φ = Ω r π Z r a r p dr dφdr . (7)We also note that the fundamental frequencies can beexpressed in elliptic integrals [18, 19]. B. Energy fluxes
Expanding the scalar field in scalar spheroidal harmon-ics and Fourier transforming, ϕ ( t, r ) = X l,m Z dω e imφ − iω t X lmω √ r + a S lm ( θ ) , (8)one can rewrite the Klein Gordon equation in the form (cid:20) d dr ∗ + V s (cid:21) X lmω ( r ) = ∆( r + a ) / T lmω , (9)where [20] V s = (cid:18) ω − m ar + a (cid:19) − ∆( r + a ) (cid:20) λ s ( r + a ) +2 M r + a ( r − M r + a ) (cid:21) − µ s ∆ r + a ,λ s is the eigenvalue of the scalar spheroidal harmonics S lm ( θ ) [4] and T lmω = − α m p S ∗ lm Z ∞−∞ dt e iωt − imφ ( t ) ( dt/dτ ) δ ( r − r ( t )) , (10)for eccentric and equatorial orbits, where S ∗ lm = S ∗ lm ( π/ X r + lmω and X ∞ lmω of the homogeneous version ofEq. (9), satisfying the boundary conditions [3, 4] X ∞ ,r + lmω ∼ e ik ∞ ,r + r ∗ as r → ∞ , r + , (11)where k + = ω − ma/ (2 M r + ), k ∞ = p ω − µ s and dr ∗ /dr = ( r + a ) / ∆. Here we take the Kerr geome-try written in Boyer-Lindquist coordinates, and r + is thelocation of the event horizon in those coordinates.Equation (9) can be solved by the Green’s functionmethod X lmω = X ∞ lmω W Z r ∗ −∞ dr ′∗ T lmω ( r ′ ) ∆( r ′ + a ) / X r + lmω ( r ′ )+ X r + lmω W Z ∞ r ∗ dr ′∗ T lmω ( r ′ ) ∆( r ′ + a ) / X ∞ lmω ( r ′ ) , where W is the Wronskian of the two homogeneous so-lutions, W = X r + lmω dX ∞ lmω /dr ∗ − X ∞ lmω dX r + lmω /dr ∗ .The inhomogeneous solution has the asymptotic format infinity as X lmω ( r → ∞ ) = e ik ∞ r ∗ W Z ∞ r + dr ′ T lmω ( r ′ ) X r + lmω ( r ′ )( r ′ + a ) / , ≡ ˜ Z ∞ lmω e ik ∞ r ∗ . (12)For eccentric and equatorial orbits, the source function T lmω has a discrete frequency spectrum and hence ˜ Z ∞ lmω is given by [17]˜ Z ∞ lmω = ∞ X k = −∞ Z ∞ lmk δ ( ω − ω mk ) , (13)where ω mk = m Ω φ + k Ω r and Z ∞ lmk = − α Ω r S ∗ lm π W Z π/ Ω r dt e iω mk t e − imφ ( t ) ( dt/dτ ) X r + lmω ( r ( t ))( r ( t ) + a ) / . Similarly, at the horizon, X lmω ( r → r + ) = e − ik + r ∗ W Z ∞ r + dr ′ T lmω ( r ′ ) X ∞ lmω ( r ′ )( r ′ + a ) / , The asymptotic behaviors of the radial function X lmω for r → ∞ and r → r + can be found by analyzing that of the potentialfunction, V s → k ∞ ( V s → k ) for r → ∞ ( r → r + ). See forexample Sec.V of Ref. [21]. ≡ ˜ Z r + lmω e − ik + r ∗ , (14)where ˜ Z r + lmω = ∞ X k = −∞ Z r + lmk δ ( ω − ω mk ) , (15)and Z r + lmk = − α Ω r S ∗ lm π W Z π/ Ω r dt e iω mk t e − imφ ( t ) ( dt/dτ ) X ∞ lmω ( r ( t ))( r ( t ) + a ) / . The scalar energy fluxes through a sphere at infinityand at the horizon are given by (cid:28) dEdt (cid:29) ∞ t = X l,m,k ω mk q ω mk − µ s | Z ∞ lmk | , (16) (cid:28) dEdt (cid:29) r + t = X l,m,k ω mk (cid:18) ω mk − ma M r + (cid:19) | Z r + lmk | . (17)In scalar-tensor theories, the equations for gravita-tional perturbations about the Kerr background are re-duced to a differential equation for the Weyl scalar Ψ [4],which satisfies the Teukolsky equation [21]. One can sep-arate the differential equation into radial and angularparts using the Fourier-harmonic decomposition,Ψ = 1( r − ia cos θ ) X l,m Z dω e imφ − iω t R lmω ( r ) S lmω ( θ ) , where S lmω ( θ ) is the spin-2 spheroidal harmonics [4]. Theseparated radial equation takes the form (cid:20) ∆ ddr (cid:18) ddr (cid:19) + V g (cid:21) R lmω ( r ) = T lmω , (18)where V g = K + 4 i ( r − M ) K ∆ − iω r − λ g ,λ g is the eigenvalue of the spin-2 spheroidal harmonics S lmω ( θ ) and T lmω is the source term derived from theenergy-momentum tensor of the point particle.From the asymptotic behaviors of the potential func-tion V g for r → ∞ and r → r + , the asymptotic forms ofthe solutions of the radial equation are derived as [21] R lmω ( r → ∞ ) ≡ ˜ Z ∞ lmω ∆ e − ik + r ∗ ,R lmω ( r → r + ) ≡ ˜ Z r + lmω r e iωr ∗ . (19)Since the source term T lmω becomes discrete in ω foreccentric and equatorial orbits, ˜ Z ∞ lmω and ˜ Z r + lmω are givenby ˜ Z ∞ lmω = ∞ X k = −∞ Z ∞ lmk δ ( ω − ω mk ) , ˜ Z r + lmω = ∞ X k = −∞ Z r + lmk δ ( ω − ω mk ) . (20) The gravitational energy fluxes through a sphere atinfinity and at the horizon are given by (cid:28) dEdt (cid:29) ∞ t = X l,m,k πω mk |Z ∞ lmk | , (21) (cid:28) dEdt (cid:29) r + t = X l,m,k α lm ( ω mk )4 πω mk |Z r + lmk | , (22)where α ℓm ( ω ) = 256(2 M r + ) k + ( k + 4˜ ǫ )( k + 16˜ ǫ ) ω |C S | , with ˜ ǫ = √ M − a / (4 M r + ) and C S is the Starobinskyconstant given by [22] |C S | = (cid:2) ( λ g + 2) + 4 aωm − a ω (cid:3) (cid:2) λ g + 36 aωm − a ω (cid:3) +(2 λ g + 3)(96 a ω − aωm ) + 144 ω ( M − a ) . C. Orbital evolution
As explained in Sec. II A, for equatorial orbits theconstants of motion of the particle, ( E , L ), depend ona semilatus rectum p and an orbital eccentricity e , i.e. E = E ( p, e ) and L = L ( p, e ). The orbital evolution of theparticle dp/dt and de/dt can be estimated by the follow-ing relation: d E dt = ∂ E ∂p dpdt + ∂ E ∂e dedt , d L dt = ∂ L ∂p dpdt + ∂ L ∂e dedt . (23)Using balance arguments for the energy and the an-gular momentum, one can compute the rate of changeof the constants of motion d E /dt and d L /dt due tothe gravitational and the scalar fluxes, which are com-puted in Sec. II B. Inverting the above equations onecan find dp/dt and de/dt . For circular orbits, dp/dt =( ∂ E /∂p ) − d E /dt since de/dt = 0. For sufficiently large p , the gravitational fluxes reduce the orbital radius andthe orbital eccentricity. However, the orbital eccentric-ity can increase ( de/dt >
0) near the last stable orbit(LSO), which is the boundary between stable and un-stable orbits [15, 17, 23–25]. Thus, while critical orbitssuch that de/dt = 0 near the LSO are possible, the con-dition dp/dt = 0 is harder to meet. Floating orbits, with dp/dt = de/dt = 0, might be possible if d E /dt = 0 and d L /dt = 0 when the gravitational fluxes are entirely com-pensated by the scalar fluxes due to superradiance for awide range of scalar-field masses [3, 4]. III. RESULTS
We have solved the above equations with an indepen-dent code and recovered to within numerical accuracy theresults reported in Refs. [3] and [4] for the case of circularorbits. Our numerical code was developed in Ref. [15] tocompute the total gravitational energy flux for eccentric
TABLE I. Location of the resonance and corresponding heightof the scalar flux h dE/dt i r + lmk normalized by α M µ for q =0 . l = m = 1 and n = 0. The point particle is orbitingon a circular geodesic. Note that the location (height) of theresonance agrees with that in Table I of Ref. [3] and Table IIof Ref. [4] around seven (four) decimal places. µ s M p/M h dE/dt i r + lmk .
35 1 . − . . . − . .
25 2 . − . . . − . .
15 3 . − . . . − . .
01 21 . − . orbits; we truncate the mode sum at l = 7 to computethe flux. When computing the scalar energy flux at thehorizon for l = m = 1, we sum over the k -modes, in orderto achieve convergence at a few decimal places.In Table I we list orbital radius at resonance in a cir-cular orbit and the corresponding height of scalar energyflux at the horizon for l = m = 1 and the n = 0 mode,and we find results from the independent code are con-sistent with that in Table I of Ref. [3] and Table II ofRef. [4]. We note that these resonances occur when fre-quencies of waves are close to the mass µ s as [26] ω res = µ s " − (cid:18) µ s Ml + 1 + n (cid:19) / , n = 0 , , . . . . (24)We also note that the height of the scalar flux at reso-nance is almost same at least for the first few overtonemodes [3]. A. Eccentric orbits
The possibility of the existence of floating orbits, i.e.,orbits for which the evolution is dominated by superra-diance was demonstrated for circular orbits [3]. It wasalso argued, but not calculated, that a small eccentricitywould not affect the results.In this section, we consider the case of eccentric orbitson the equatorial plane of the BH to examine if a floatingis possible for massive scalar field. As a code check tocompute the scalar energy flux for eccentric orbits, wehave compared the energy flux with the one at the leadingpost-Newtonian order in Ref. [27], and found our resultsare consistent for the case of eccentric orbits and themassless scalar field.In Fig. 1, we compare the total gravitational energyflux with the modal scalar energy flux at the horizon for l = m = 1, n = 0, and k = − , q = 0 . µ s M = 0 . α = 0 .
01 and e = 0 .
1. In the figure, wefind three peaks in the scalar energy flux at the horizoncorresponding to k = −
1, 0 and 1 modes from left toright, and find that floating would be possible for k = 0 -18 -16 -14 -12 -10 -8 -6 -4 -2
5 10 15 20 25 30 35 40 < d E / d t > p/M µ s M=0.01, α =0.01,e=0.1
01 and e = 0 . p/M . For these parameters,floating orbits are possible for k = 0 and k = 1 modes. Thelocation and height of the peak and α crit for l = m = 1 and n = 0 when q = 0 . µ s M = 0 .
01 and e = 0 . and 1 modes. In Table II, we also show the location,the height of the peak and the critical value of α for afloating orbit to be possible. We fix l = m = 1, n =0, q = 0 .
99 but vary µ s M , the k -mode and the orbitaleccentricity. We note that α crit in the table is smallerthan current bounds on α for most of cases, and hencethe orbital eccentricity would not reduce the resonanceeffect significantly. B. Finite-size effects on resonance
In this section, we would like to estimate finite-sizeeffects on these superradiant resonances. A proper treat-ment is too complex and outside the scope of our work.Instead, we will estimate finite-size effects by modelingan extended body as made up of a collection of point-like noninteracting particles. Such simplification is notnew [29–32]; however, one needs to exercise extra care inthis context. If the swarm of particles is to describe acompact object, then one needs to impose that the ra-dial and angular extent is kept finite at all times. Thereare at least two different ways to realize this idea andrestriction.In Sec. III B 1, we deal with particles on a circular or-bit. These particles mimic a body which is extended inthe angular direction but pointlike in the radial direc-tion. In Sec. III B 2, we drop the assumption that theparticles composing the body are on a circular orbit andconsider instead particles in eccentric orbits around theequatorial plane of a Kerr BH. Generically, the properdistance between these two particles varies substantially,and the swarm of particles would not mimic a compact,rigid body. However, for particles with the same funda-mental frequencies and (slightly) different orbital param-
TABLE II. Location of the resonance and corresponding height of the scalar flux h dE/dt i r + lmk normalized by α M µ for q = 0 . l = m = 1, n = 0, and µ s M = 0 .
01 (left) and µ s M = 0 . e = 0 . , . , . , . α ≥ α crit . Note that, for scalar-tensor theories, α crit is well below current observational constraintsfor µ s considered in the table [28], for eccentricities e = 0 . , . µ s M = 0 . e = 0 . µ s M = 0 . e = 0 . k p/M h dE/dt i r + lmk α crit p/M h dE/dt i r + lmk α crit − . − . × − . × − . − . × − .
10 21 . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . µ s M = 0 . e = 0 . µ s M = 0 . e = 0 . k p/M h dE/dt i r + lmk α crit p/M h dE/dt i r + lmk α crit − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − µ s M = 0 . e = 0 . µ s M = 0 . e = 0 . k p/M h dE/dt i r + lmk α crit p/M h dE/dt i r + lmk α crit − . − . × − . . − . × − . − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − µ s M = 0 . e = 0 . µ s M = 0 . e = 0 . k p/M h dE/dt i r + lmk α crit p/M h dE/dt i r + lmk α crit − . − . × − . . − . × − . − . − . × − . × − . − . × − . − . − . × − . × − . − . × − .
40 11 . − . × − . × − . − . × − .
21 17 . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − . − . × − . × − eters, i.e. in “isofrequency pairing” [33], then the properdistance between the particles is finite and varies onlyslightly and in a controlled way as time goes by. Theseare therefore a good model also for an extended body.
1. Particles in a circular orbit
Let us first start with a body which is pointlike inthe radial direction and composed of 2 N + 1 identicalpointlike particles of mass µ/ (2 N + 1) in a circular orbitaround a Kerr BH of mass M . We set the orbit of the j th particle as φ j ( t ) = Ω φ t + j N + 1 δ, j = 0 , ± , ± , ..., ± N , (25)where Ω φ = √ M / ( a √ M + r / ). We choose δ =arcsin( µ/r ), in order to model the typical size of astellar-mass compact star in a circular orbit around aKerr BH. Note in the limit N → ∞ the group of theparticles represents a circular arc with the angular size δ . (For the case δ = 2 π , i.e. a ring, see Sec. 6.1 of Part IIin Ref. [29].)The source term of the field equations takes the form T ℓmω = T (0) ℓmω f ( m, N ) , (26) -12 -11 -10
1 100 10000 1e+06 1e+08 1e+10 - f ( m , N ) N δ =10 -5 m=1m=2m=3 10 -16 -15 -14
1 100 10000 1e+06 1e+08 1e+10 - f ( m , N ) N δ =10 -7 m=1m=2m=3 FIG. 2. Plots for 1 − f ( m, N ) when δ = 10 − and 10 − fromleft to right. Finite-size effects along the φ direction are, forall purposes, negligible since 1 − f ( m, N ) < − for m ≤ N ≤ and δ ≤ − . Note that f ( m, N ) approaches 1 ifone of m , N , and δ approaches zero. where f ( m, N ) = 12 N + 1 j = N X j = − N e i m j N +1 δ , = 12 N + 1 j = N X j =1 cos (cid:18) j N + 1 m δ (cid:19) , ≤ , (27)and T (0) ℓmω is the source term of a single particle.The condition f ( m, N ) ≤ δ =arcsin[( µ/M )( M/r )] ≪
1. Figure 2 shows numericalvalues for 1 − f ( m, N ) for δ = 10 − , − . We find that1 − f ( m, N ) < − for m ≤ N ≤ and δ ≤ − .Moreover, the cancellation affects both the scalar andgravitational channels equally. Thus, the suppression isunder control, and this type of finite-size effects cannotsuppress floating or the resonances.
2. Particles in isofrequency orbits
Let us now consider an object which is of finite ex-tent in both the radial and azimuthal directions. As asimple toy model, take two particles which are in isofre-quency pairing [33] for eccentric orbits around the equa-torial plane of a Kerr BH.In an isofrequency pairing situation, two different or-bits can have the same fundamental frequencies near thelast stable orbit, satisfying the conditions, r (1) ( t ) = r (2) ( t ) , φ (1) ( t ) = φ (2) ( t ) , Ω (1) r = Ω (2) r , Ω (1) φ = Ω (2) φ , (28)where Ω ( j ) r (Ω ( j ) φ ) is the radial (azimuthal) frequency ofthe j th particle and j = 1 , r in the (Ω φ , e ) parameterspace when q = 0 .
9, where e is the eccentricity of the e M Ω φ q=0.9 M Ω r FIG. 3. Contours for Ω r in the (Ω φ , e ) parameter space for q =0 .
9. There seems to exist isofrequency orbits for M Ω φ & . M Ω φ = 0 .
235 intersects the M Ω r =0 .
015 curve at e ∼ . .
58; the two intersections aretherefore an example of isofrequency orbits. More precisevalues of orbital parameters for isofrequency orbits are shownin Table III. orbit defined through r = p/ (1 + e cos χ ) and χ is therelativistic anomaly parameter. (See also Figs. 1 and3 in Ref. [33].) This figure suggests that one can findisofrequency orbits of which the orbital parameters arevery similar, p (1) ∼ p (2) and e (1) ∼ e (2) , in the regionclose to the curve along which the Jacobian matrix ofthe transformation ( p, e ) ↔ (Ω r , Ω φ ) becomes singular.Since r ( j ) ( t ) = r ( j ) ( t + n T r ) , j = 1 , ,φ ( j ) ( t ) = Ω φ t + ∆ φ ( j ) ( t ) , (29)where T r = 2 π/ Ω r and ∆ φ ( j ) ( t ) is the oscillating part of φ ( j ) ( t ) with the period T r , r (1) ( t ) − r (2) ( t ) and φ (1) ( t ) − φ (2) ( t ) might be constant in the time-averaged sense, ifisofrequency orbits have very similar orbital parameters.In Table III, we show examples of isofrequency orbitsfor q = 0 . , . q = 0 .
99. In the table, we choose thedifference of orbital parameters for two particles to be assmall as 1 − p (2) /p (1) ∼ − and 1 − e (2) /e (1) ∼ − .We still need to show that the distance between the twoparticles on these orbits is bounded from above by valuesthat can mimic a compact star. This study is shownin Fig. 4, where we show the distance between the twoparticles as a function of time. This distance in fact islimited to values bounded from above by the length scaleof mass µ = ( µ/M ) M ∼ − M . Thus, isofrequencyorbits can be used to investigate some of finite-size effectsfrom two particles.In Table IV, we compute the height of the resonantscalar flux when q = 0 . l = m = 1, and k = 0. We findthat the relative difference in the flux for two particlesin isofrequency orbits, (cid:10) dE (1) /dt (cid:11) r + lmk and (cid:10) dE (2) /dt (cid:11) r + lmk ,is comparable to that in the orbital parameters p and TABLE III. Examples of isofrequency orbits for q = 0 . , . .
99. Note 1 − p (2) /p (1) ∼ − , 1 − e (2) /e (1) ∼ − ,1 − Ω (2) φ / Ω (1) φ ∼ − and 1 − Ω (2) r / Ω (1) r ∼ − . Note also that the high precision of the parameter values is necessary for theorbital frequencies to agree in the 13 digits. This order of accuracy is necessary because, near the separatrix, small changes inthe orbital parameters can result in comparatively large changes in the frequencies. The total energy flux refers to expression,and intends to describe the total flux from a particle with fixed mass (and equal to the masses of the individual particles 1 and2). Particle p e Ω φ Ω r q = 0 . . . . . . . . . q = 0 . . . . . . . . . q = 0 .
99 1 1 . . . . . . . . q = 0 .
99 1 1 . . . . . . . . q = 0 .
99 1 1 . . . . . . . . q = 0 .
99 1 1 . . . . . . . . q = 0 .
99 1 1 . . . . . . . . ⋅ -6 ⋅ -6 ⋅ -6 ⋅ -6 ⋅ -6 ⋅ -6
0 200 400 600 800 1000 | → x ( t )- → x ( t ) | t/Mq=0.99, p/M~1.65103, e~0.25698 -1.5 ⋅ -6 -1.0 ⋅ -6 -5.0 ⋅ -7 ⋅ ⋅ -7
0 200 400 600 800 1000 r ( t )-r ( t ) t/Mq=0.99, p/M~1.65103, e~0.25698 -4.0 ⋅ -6 -3.0 ⋅ -6 -2.0 ⋅ -6 -1.0 ⋅ -6 ⋅ ⋅ -6 ⋅ -6 ⋅ -6 ⋅ -6 ⋅ -6
0 200 400 600 800 1000 φ ( t )- φ ( t ) t/Mq=0.99, p/M~1.65103, e~0.25698 FIG. 4. Differences in the evolution of x ( t ) (left), r ( t ) (middle), and φ ( t ) (right), with periods T r = 2 π/ Ω r , for q = 0 .
99 inTable III. A bounded value of 0 < | x (1) ( t ) − x (2) ( t ) | / − M for the difference in position between both particles indicatesthat the two particles stay in the length scale of mass µ ∼ − M . e , ∼ − . We define two notions of total energyflux, intended to describe an object with fixed totalmass. The “averaged flux” is computed as the aver-age of the fluxes of the first and second particles, i.e., (cid:16)(cid:10) dE (1) /dt (cid:11) r + lmk + (cid:10) dE (2) /dt (cid:11) r + lmk (cid:17) /
2. The “total” flux,on the other hand, is computed using the source term T ℓmω = T (1) ℓmω + T (2) ℓmω , (30)where T ( j ) ℓmω is the source term of the j th particle with j = 1 ,
2. The relative difference in the scalar flux forthe total and averaged flux is around 10 − or better,which is ∼ (1 − e (2) /e (1) ) and might be understood fromEq. (17).Another issue concerns the extension to N particlesin isofrequency orbits: there is no guarantee that isofre-quency orbits exist for N -particle systems when N ≥ N -particle systems by consideringΩ ( i ) φ = Ω ( j ) φ , Ω ( i ) r ∼ Ω ( j ) r , p ( i ) ∼ p ( j ) , and e ( i ) ∼ e ( j ) with0 < | x ( i ) ( t ) − x ( j ) ( t ) | / − M where i, j = 1 , , ..., N ,which can be found not only in the region near the LSObut also farther away.In fact, one can find such quasi-isofrequency orbits when p (1) ≤ p ≤ p (2) . If we define a semilatus rectum p as p ( i,j ) = p (1) + 2 j − i ( p (2) − p (1) ) , (31)where i is a positive integer and 1 ≤ j ≤ i − , wecan find corresponding orbital eccentricity e ( i,j ) whichgives e (1) ≤ e ( i,j ) ≤ e (2) , 1 − e ( i,j ) /e ( i ′ ,j ′ ) . − ,1 − Ω ( i,j ) φ / Ω ( i ′ ,j ′ ) φ . − and 1 − Ω ( i,j ) r / Ω ( i ′ ,j ′ ) r ∼ − .If we define semilatus rectum p as p (0 , ≡ p (1) when TABLE IV. Resonance and corresponding height of the scalar flux h dE/dt i r + lmk normalized by α M µ for q = 0 .
99 and l = m = 1, and k = 0. Orbital parameters p and e are same as those in Table III. The relative difference in the scalar fluxes fortwo particles in isofrequency orbits, D dE (1) /dt E r + lmk and D dE (2) /dt E r + lmk , is comparable to 1 − p (2) /p (1) and 1 − e (2) /e (1) . Theaveraged scalar flux is the average of the fluxes of the first and the second particles. The total scalar flux is computed using thesource term (cid:16) T (1) ℓmω + T (2) ℓmω (cid:17) /
2, where T ( j ) ℓmω is the source term of the j th particle with j = 1 ,
2. The relative difference betweenthe scalar flux for the total and averaged flux is around 10 − or better, which is ∼ (1 − e (2) /e (1) ) and might be understoodfrom Eq. (17). µ s M Particle p/M e h dE/dt i r + lmk . . . − . × − . . − . × − Averaged − − − . × − Total − − − . × − . . . − . × − . . − . × − Averaged − − − . × − Total − − − . × − . . . − . × − . . − . × − Averaged − − − . × − Total − − − . × − . . . − . × − . . − . × − Averaged − − − . × − Total − − − . × − . . . − . × − . . − . × − Averaged − − − . × − Total − − − . × − ( i, j ) = (0 ,
1) and p (0 , ≡ p (2) when ( i, j ) = (0 , ≤ i ≤ i max is given as N = 2 i max + 1. We note, for the N particle, the dif-ference in semilatus rectums for neighboring particles isgiven as ( p (2) − p (1) ) / ( N − N particle using the source term T ℓmω = 1 N i max X i =0 X j T ( i,j ) ℓmω , (32)where T ( i,j ) ℓmω is the source term of the particle with theorbital parameters p ( i,j ) and e ( i,j ) .In Fig. 5, the absolute values of the relative differencein the scalar flux for the N particle and that for total areshown as a function of N when q = 0 . l = m = 1, k =0 and 0 ≤ i max ≤
10, i.e., 2 ≤ N ≤ p (1 , and e (1 , , which is the center of the N particle alongthe radial direction, is ∼ − or smaller. Since thisdifference is smaller than the relative difference in theorbital parameters of the N particle (1 − e (2) /e (1) ) , thefinite-size effects along the radial direction are not enoughto suppress floating or the resonance. -14 -13 -12 -11 -10 -9
1 10 100 1000 - < d E d t > r + ( N ) / < d E d t > r + ( N = ) N µ s M=0.3524737301 µ s M=0.3578874552 µ s M=0.3651257594 µ s M=0.3738869266
FIG. 5. The absolute values of the relative difference in thescalar flux for the N particle and that for total are shownas a function of N when q = 0 . l = m = 1, k = 0 and0 ≤ i max ≤
10, i.e., 2 ≤ N ≤ IV. CONCLUSION AND DISCUSSION
We have considered a stellar-mass compact star or-biting a supermassive BH in scalar-tensor theories. InRefs. [3, 4] it was shown that a floating, circular orbitis possible due to resonances excited when the orbitalperiod becomes comparable to the mass of the scalarfield. In this paper, we have extended the analysis inRefs. [3, 4] to eccentric orbits and to a group of testparticles to investigate whether resonances due to thecoupling of the scalar field to matter in EMRIs are af-fected by the orbital eccentricity and finite-size effects ofthe orbiting star, modeled by the group of particles. Wehave found that these effects do not reduce the resonancesignificantly.From Tables I and II, it is found that, for a given µ s ,scalar fluxes for the case of a circular orbit are compa-rable to those for the case of a low eccentric orbit with k = 0. As the orbital eccentricity and | k | increase, how-ever, there appears an order of magnitude difference influxes for circular and eccentric orbits for a given µ s .This might be understood as follows: the scalar energyfluxes are computed through Eqs. (16) and (17), whichrequire the Wronskian and the integration in time for theasymptotic amplitudes Z ∞ lmk and Z r + lmk . Since the Wron-skian is computed from the two homogeneous solutions ofthe Klein-Gordon equation and is independent of the ra-dial coordinate, its value for an eccentric orbit for a givenset of q , l , m , n , µ s and resonant frequency is the sameas that of a circular orbit. Thus, the main difference inthe scalar flux calculation between circular and eccentricorbits at resonances for a given µ s lies in the integrationto find Z ∞ lmk and Z r + lmk . Noting the integration becomeszero for circular orbits when k = 0, we may find that theintegration for eccentric orbits when k = 0 is an orderof magnitude smaller than that for circular orbits when k = 0 for a given µ s . The same argument carries overto inclined orbits, and one therefore expects that orbitalinclination does not reduce the resonance significantly.We have also considered a collection of point particles,which were intended to mimic extended bodies and finite-size effects on the resonance. We found that finite-sizeeffects along the azimuthal direction do induce a phasecancellation, but the cancellation is very small and typ-ically unimportant. Since the cancellation affects boththe scalar and the gravitational flux equally, we expectthat finite-size effects along the azimuthal direction arenot enough to suppress floating or resonance. We thenconsidered particles in quasi-isofrequency orbits , that arean extension of isofrequency orbits [33], to take into ac-count finite-size effects along the radial and azimuthaldirections. Again we found that this type of finite-sizeeffects is very small and is below the relative differencein orbital parameters of particles. Thus we expect thatfinite-size effects modeled by a collection of particles arenot enough to suppress floating or resonance.The spin of the orbiting particle is another ingredientthat should be considered to investigate finite-size effects on the resonance. The orbits of a spinning particle can bechaotic [34–41], and one might not be able to deal withthe system in the frequency domain. At linear order inthe spin of the particle, however, it is possible to definefrequencies of the orbit and spin precession for simplecases [42, 43]. For generic orbits, it is not clear if onecan define frequencies of the orbit and spin precession,although there still exist three constants of motion [44].Reference[45] suggests solving the evolution of the orbitand the spin of the particle using the osculating geodesicmethod [46], making it possible to define the frequenciesof the orbit and spin precession at each geodesic [45]. Inthis case, however, the frequencies of the orbit and spinprecession oscillate in time, and the time scale of theoscillation is comparable to that of orbits. If the ampli-tudes of the oscillation in the frequencies are larger thanthe width of the resonance in the frequency, floating maynot be possible or may not last long enough to distinguishit from nonfloating orbits.In summary, we computed the scalar and gravitationalenergy flux to investigate whether floating is possible foreccentric orbits or for extended bodies. These fluxes arethe time-averaged dissipative part of the first-order self-force that would induce deviations from the geodesic mo-tion at the first order in the mass ratio of the system [47–49]. The conservative self-force is the remaining part ofthe self-force which is free from dissipation in the system.When one considers the motion taking into account thefirst-order conservative self-force, one may define orbitalfrequencies in the “conservative” effective spacetime [50].Those frequencies might not coincide with resonant fre-quencies even if orbital frequencies in geodesic motioncoincide with a resonant frequency. It is therefore im-portant to solve the self-force equation in scalar-tensortheories [51] and to compute the self-force to investigatethe stability of floating orbits. These issues are left forfuture works. ACKNOWLEDGMENTS
We would like to thank Paolo Pani for useful discus-sions. We acknowledge financial support provided underthe European Union’s H2020 ERC Consolidator Grant“Matter and strong-field gravity: New frontiers in Ein-stein’s theory” Grant No. MaGRaTh–646597. Researchat Perimeter Institute is supported by the Government ofCanada through Industry Canada and by the Provinceof Ontario through the Ministry of Economic Develop-ment and Innovation. This project has received fundingfrom the European Union’s Horizon 2020 research andinnovation program under the Marie Sklodowska-CurieGrant No 690904. The authors thankfully acknowledgethe computer resources, technical expertise and assis-tance provided by CENTRA/IST and by the Yukawa In-stitute Computer Facility. Computations were performedat the clusters “Baltasar-Sete-S´ois” and at the YukawaInstitute Computer Facility and were supported by the0MaGRaTh–646597 ERC Consolidator Grant. [1] E. Berti et al. , Class. Quant. Grav. , 243001 (2015)doi:10.1088/0264-9381/32/24/243001 [arXiv:1501.07274[gr-qc]].[2] J. Maga˜na, T. Matos, V. Robles and A. Su´arez, Pro-ceedings of the XIII Mexican Workshop on Particles andFields; arXiv:1201.6107 [astro-ph.CO].[3] V. Cardoso, S. Chakrabarti, P. Pani, E. Berti andL. Gualtieri, Phys. Rev. Lett. , 241101 (2011)doi:10.1103/PhysRevLett.107.241101 [arXiv:1109.6021[gr-qc]].[4] N. Yunes, P. Pani and V. Cardoso, Phys. Rev.D , 102003 (2012) doi:10.1103/PhysRevD.85.102003[arXiv:1112.3351 [gr-qc]].[5] E. Berti, V. Cardoso, L. Gualtieri, M. Horbatschand U. Sperhake, Phys. Rev. D , 124020 (2013)doi:10.1103/PhysRevD.87.124020 [arXiv:1304.2836 [gr-qc]].[6] V. Cardoso, I. P. Carucci, P. Pani andT. P. Sotiriou, Phys. Rev. Lett. , 111101 (2013)doi:10.1103/PhysRevLett.111.111101 [arXiv:1308.6587[gr-qc]].[7] R. Brito, V. Cardoso and P. Pani, Lect. NotesPhys. , 1 (2015) doi:10.1007/978-3-319-19000-6[arXiv:1501.06570 [gr-qc]].[8] C. Hopman and T. Alexander, Astrophys. J. , 362(2005).[9] M. Shibata, Prog. Theor. Phys. , 595 (1993).[10] M. Shibata, M. Sasaki, H. Tagoshi and T. Tanaka, Phys.Rev. D , 1646 (1995).[11] S. A. Hughes, Phys. Rev. D , 084004 (2000).[12] S. A. Hughes, Phys. Rev. D , 064004 (2001).[13] N. Sago T. Tanaka, W. Hikida, K. Ganz and H. Nakano,Prog. Theor. Phys. , 873 (2006).[14] K. Ganz, W. Hikida, H. Nakano, N. Sago and T. Tanaka,Prog. Theor. Phys. , 1041 (2007).[15] R. Fujita, W. Hikida and H. Tagoshi, Prog. Theor. Phys. , 843 (2009).[16] N. Sago and R. Fujita, Prog. Theor. Exp. Phys. ,073E03 (2015).[17] K. Glampedakis and D. Kennefick, Phys. Rev. D 66 ,044002 (2002). arXiv:0203086 [gr-qc].[18] W. Schmidt, Class. Quant. Grav. , 2743 (2002).[19] R. Fujita and W. Hikida, Class. Quant. Grav. , 135002(2009).[20] A. Ohashi H. Tagoshi and M. Sasaki, Prog. Theor. Phys. , 713 (1996).[21] S. A. Teukolsky, Astrophys. J. , 635 (1973).[22] S. A. Teukolsky and W. H. Press, Astrophys. J. , 443(1974).[23] T. Tanaka, M. Shibata, M. Sasaki, H. Tagoshi and T. Nakamura, Prog. Theor. Phys. , 65 (1993).[24] C. Cutler, D. Kennefick and E. Poisson, Phys. Rev. D50 , 3816 (1994).[25] D. Kennefick, Phys. Rev.
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